On FreeBMD, you won't find any parents' names, except for some of the later births, when you will find the mother's maiden names.
There is nowhere to enter parents' names in the search.
If you found three John Smiths, you would not know which one was yours unless you already knew the mother's maiden name and it was a birth after 1912.
On GRO Services, which only does births and deaths (not marriages), you will find the mother's maiden name for births, and the age for deaths. Births only go up to 1917, and deaths only go up to 1957.